Fundamentals of physics

1334 Pages·2007·21.711 MB·English

by  Jearl Walker, Robert Resnick, David Halliday

PP UU ZZ ZZ LL EE RR
For thousands of years the spinning
Earth provided a natural standard for our
measurements of time. However, since
1972 we have added more than 20 “leap
seconds” to our clocks to keep them
synchronized to the Earth. Why are such
adjustments needed? What does it take
to be a good standard? (Don Mason/The
Stock Market andNASA)
c h a p t e r
Physics and Measurement
Chapter Outline
1.1 Standards of Length, Mass, and 1.5 Conversion of Units
Time 1.6 Estimates and Order-of-Magnitude
1.2 The Building Blocks of Matter Calculations
1.3 Density 1.7 Significant Figures
1.4 Dimensional Analysis
2
L
ike all other sciences, physics is based on experimental observations and quan-
titative measurements. The main objective of physics is to find the limited num-
ber of fundamental laws that govern natural phenomena and to use them to
develop theories that can predict the results of future experiments. The funda-
mental laws used in developing theories are expressed in the language of mathe-
matics, the tool that provides a bridge between theory and experiment.
When a discrepancy between theory and experiment arises, new theories must
be formulated to remove the discrepancy. Many times a theory is satisfactory only
under limited conditions; a more general theory might be satisfactory without
such limitations. For example, the laws of motion discovered by Isaac Newton
(1642–1727) in the 17th century accurately describe the motion of bodies at nor-
mal speeds but do not apply to objects moving at speeds comparable with the
speed of light. In contrast, the special theory of relativity developed by Albert Ein-
stein (1879–1955) in the early 1900s gives the same results as Newton’s laws at low
speeds but also correctly describes motion at speeds approaching the speed of
light. Hence, Einstein’s is a more general theory of motion.
Classical physics, which means all of the physics developed before 1900, in-
cludes the theories, concepts, laws, and experiments in classical mechanics, ther-
modynamics, and electromagnetism.
Important contributions to classical physics were provided by Newton, who de-
veloped classical mechanics as a systematic theory and was one of the originators
of calculus as a mathematical tool. Major developments in mechanics continued in
the 18th century, but the fields of thermodynamics and electricity and magnetism
were not developed until the latter part of the 19th century, principally because
before that time the apparatus for controlled experiments was either too crude or
unavailable.
A new era in physics, usually referred to as modern physics, began near the end
of the 19th century. Modern physics developed mainly because of the discovery
that many physical phenomena could not be explained by classical physics. The
two most important developments in modern physics were the theories of relativity
and quantum mechanics. Einstein’s theory of relativity revolutionized the tradi-
tional concepts of space, time, and energy; quantum mechanics, which applies to
both the microscopic and macroscopic worlds, was originally formulated by a num-
ber of distinguished scientists to provide descriptions of physical phenomena at
the atomic level.
Scientists constantly work at improving our understanding of phenomena and
fundamental laws, and new discoveries are made every day. In many research
areas, a great deal of overlap exists between physics, chemistry, geology, and
biology, as well as engineering. Some of the most notable developments are
(1) numerous space missions and the landing of astronauts on the Moon,
(2) microcircuitry and high-speed computers, and (3) sophisticated imaging tech-
niques used in scientific research and medicine. The impact such developments
and discoveries have had on our society has indeed been great, and it is very likely
that future discoveries and developments will be just as exciting and challenging
and of great benefit to humanity.
1.1 STANDARDS OF LENGTH, MASS, AND TIME
The laws of physics are expressed in terms of basic quantities that require a clear def-
inition. In mechanics, the three basic quantities are length (L), mass (M), and time
(T). All other quantities in mechanics can be expressed in terms of these three.
3
4 CHAPTER 1 Physics and Measurements
If we are to report the results of a measurement to someone who wishes to re-
produce this measurement, a standardmust be defined. It would be meaningless if
a visitor from another planet were to talk to us about a length of 8 “glitches” if we
do not know the meaning of the unit glitch. On the other hand, if someone famil-
iar with our system of measurement reports that a wall is 2 meters high and our
unit of length is defined to be 1 meter, we know that the height of the wall is twice
our basic length unit. Likewise, if we are told that a person has a mass of 75 kilo-
grams and our unit of mass is defined to be 1 kilogram, then that person is 75
times as massive as our basic unit.1Whatever is chosen as a standard must be read-
ily accessible and possess some property that can be measured reliably—measure-
ments taken by different people in different places must yield the same result.
In 1960, an international committee established a set of standards for length,
mass, and other basic quantities. The system established is an adaptation of the
metric system, and it is called the SI system of units. (The abbreviation SI comes
from the system’s French name “Système International.”) In this system, the units
of length, mass, and time are the meter, kilogram, and second, respectively. Other
SI standards established by the committee are those for temperature (the kelvin),
electric current (the ampere), luminous intensity (the candela), and the amount of
substance (the mole). In our study of mechanics we shall be concerned only with
the units of length, mass, and time.
Length
In A.D. 1120 the king of England decreed that the standard of length in his coun-
try would be named the yardand would be precisely equal to the distance from the
tip of his nose to the end of his outstretched arm. Similarly, the original standard
for the foot adopted by the French was the length of the royal foot of King Louis
XIV. This standard prevailed until 1799, when the legal standard of length in
France became the meter,defined as one ten-millionth the distance from the equa-
tor to the North Pole along one particular longitudinal line that passes through
Paris.
Many other systems for measuring length have been developed over the years,
but the advantages of the French system have caused it to prevail in almost all
countries and in scientific circles everywhere. As recently as 1960, the length of the
meter was defined as the distance between two lines on a specific platinum–
iridium bar stored under controlled conditions in France. This standard was aban-
doned for several reasons, a principal one being that the limited accuracy with
which the separation between the lines on the bar can be determined does not
meet the current requirements of science and technology. In the 1960s and 1970s,
the meter was defined as 1 650 763.73 wavelengths of orange-red light emitted
from a krypton-86 lamp. However, in October 1983, the meter (m) was redefined
as the distance traveled by light in vacuum during a time of 1/299 792 458
second. In effect, this latest definition establishes that the speed of light in vac-
uum is precisely 299 792 458m per second.
Table 1.1 lists approximate values of some measured lengths.
1 The need for assigning numerical values to various measured physical quantities was expressed by
Lord Kelvin (William Thomson) as follows: “I often say that when you can measure what you are speak-
ing about, and express it in numbers, you should know something about it, but when you cannot ex-
press it in numbers, your knowledge is of a meagre and unsatisfactory kind. It may be the beginning of
knowledge but you have scarcely in your thoughts advanced to the state of science.”
1.1 Standards of Length, Mass, and Time 5
TABLE 1.1 Approximate Values of Some Measured Lengths
Length (m)
Distance from the Earth to most remote known quasar 1.4(cid:1)1026
Distance from the Earth to most remote known normal galaxies 9(cid:1)1025
Distance from the Earth to nearest large galaxy
(M 31, the Andromeda galaxy) 2(cid:1)1022
Distance from the Sun to nearest star (Proxima Centauri) 4(cid:1)1016
One lightyear 9.46(cid:1)1015
Mean orbit radius of the Earth about the Sun 1.50(cid:1)1011
Mean distance from the Earth to the Moon 3.84(cid:1)108
Distance from the equator to the North Pole 1.00(cid:1)107
Mean radius of the Earth 6.37(cid:1)106
Typical altitude (above the surface) of a satellite orbiting the Earth 2(cid:1)105
Length of a football field 9.1(cid:1)101
Length of a housefly 5(cid:1)10(cid:2)3
Size of smallest dust particles (cid:1)10(cid:2)4
Size of cells of most living organisms (cid:1)10(cid:2)5
Diameter of a hydrogen atom (cid:1)10(cid:2)10
Diameter of an atomic nucleus (cid:1)10(cid:2)14
Diameter of a proton (cid:1)10(cid:2)15
Mass
The basic SI unit of mass, the kilogram (kg), is defined as the mass of a spe- web
cific platinum–iridium alloy cylinder kept at the International Bureau of Visit the Bureau at www.bipm.fror the
Weights and Measures at Sèvres, France.This mass standard was established in National Institute of Standards at
www.NIST.gov
1887 and has not been changed since that time because platinum–iridium is an
unusually stable alloy (Fig. 1.1a). A duplicate of the Sèvres cylinder is kept at the
National Institute of Standards and Technology (NIST) in Gaithersburg, Maryland.
Table 1.2 lists approximate values of the masses of various objects.
TABLE 1.2
Time Masses of Various Bodies
(Approximate Values)
Before 1960, the standard of time was defined in terms of the mean solar dayfor the
year 1900.2 The mean solar second was originally defined as (1)(1)(1) of a mean Body Mass (kg)
60 60 24
solar day. The rotation of the Earth is now known to vary slightly with time, how-
ever, and therefore this motion is not a good one to use for defining a standard. Visible (cid:1)1052
Universe
In 1967, consequently, the second was redefined to take advantage of the high
Milky Way 7(cid:1)1041
precision obtainable in a device known as an atomic clock(Fig. 1.1b). In this device,
galaxy
the frequencies associated with certain atomic transitions can be measured to a
Sun 1.99(cid:1)1030
precision of one part in 1012. This is equivalent to an uncertainty of less than one Earth 5.98(cid:1)1024
second every 30 000 years. Thus, in 1967 the SI unit of time, the second, was rede- Moon 7.36(cid:1)1022
fined using the characteristic frequency of a particular kind of cesium atom as the Horse (cid:1)103
“reference clock.” The basic SI unit of time, the second (s), is defined as 9 192 Human (cid:1)102
631 770 times the period of vibration of radiation from the cesium-133 Frog (cid:1)10(cid:2)1
atom.3 To keep these atomic clocks—and therefore all common clocks and Mosquito (cid:1)10(cid:2)5
Bacterium (cid:1)10(cid:2)15
Hydrogen 1.67(cid:1)10(cid:2)27
2 One solar day is the time interval between successive appearances of the Sun at the highest point it atom
reaches in the sky each day. Electron 9.11(cid:1)10(cid:2)31
3 Periodis defined as the time interval needed for one complete vibration.
6 CHAPTER 1 Physics and Measurements
Figure 1.1 (Top)The National Standard Kilogram No.
20, an accurate copy of the International Standard Kilo-
gram kept at Sèvres, France, is housed under a double bell
jar in a vault at the National Institute of Standards and
Technology (NIST). (Bottom)The primary frequency stan-
dard (an atomic clock) at the NIST. This device keeps
time with an accuracy of about 3 millionths of a second
per year. (Courtesy of National Institute of Standards and Technology,
U.S. Department of Commerce)
watches that are set to them—synchronized, it has sometimes been necessary to
add leap seconds to our clocks. This is not a new idea. In 46 B.C. Julius Caesar be-
gan the practice of adding extra days to the calendar during leap years so that the
seasons occurred at about the same date each year.
Since Einstein’s discovery of the linkage between space and time, precise mea-
surement of time intervals requires that we know both the state of motion of the
clock used to measure the interval and, in some cases, the location of the clock as
well. Otherwise, for example, global positioning system satellites might be unable
to pinpoint your location with sufficient accuracy, should you need rescuing.
Approximate values of time intervals are presented in Table 1.3.
In addition to SI, another system of units, the British engineering system (some-
times called the conventional system), is still used in the United States despite accep-
tance of SI by the rest of the world. In this system, the units of length, mass, and
1.1 Standards of Length, Mass, and Time 7
TABLE 1.3 Approximate Values of Some Time Intervals
Interval (s)
Age of the Universe 5(cid:1)1017
Age of the Earth 1.3(cid:1)1017
Average age of a college student 6.3(cid:1)108
One year 3.16(cid:1)107
One day (time for one rotation of the Earth about its axis) 8.64(cid:1)104
Time between normal heartbeats 8(cid:1)10(cid:2)1
Period of audible sound waves (cid:1)10(cid:2)3
Period of typical radio waves (cid:1)10(cid:2)6
Period of vibration of an atom in a solid (cid:1)10(cid:2)13
Period of visible light waves (cid:1)10(cid:2)15
Duration of a nuclear collision (cid:1)10(cid:2)22
Time for light to cross a proton (cid:1)10(cid:2)24
time are the foot (ft), slug, and second, respectively. In this text we shall use SI
units because they are almost universally accepted in science and industry. We
shall make some limited use of British engineering units in the study of classical
mechanics.
In addition to the basic SI units of meter, kilogram, and second, we can also
use other units, such as millimeters and nanoseconds, where the prefixes milli-and
nano- denote various powers of ten. Some of the most frequently used prefixes
for the various powers of ten and their abbreviations are listed in Table 1.4. For
TABLE 1.4 Prefixes for SIUnits
Power Prefix Abbreviation
10(cid:2)24 yocto y
10(cid:2)21 zepto z
10(cid:2)18 atto a
10(cid:2)15 femto f
10(cid:2)12 pico p
10(cid:2)9 nano n
10(cid:2)6 micro (cid:3)
10(cid:2)3 milli m
10(cid:2)2 centi c
10(cid:2)1 deci d
101 deka da
103 kilo k
106 mega M
109 giga G
1012 tera T
1015 peta P
1018 exa E
1021 zetta Z
1024 yotta Y
8 CHAPTER 1 Physics and Measurements
example, 10(cid:2)3m is equivalent to 1 millimeter (mm), and 103m corresponds
to 1 kilometer (km). Likewise, 1kg is 103grams (g), and 1 megavolt (MV) is
106volts (V).
u u 1.2 THE BUILDING BLOCKS OF MATTER
A 1-kg cube of solid gold has a length of 3.73cm on a side. Is this cube nothing
d
but wall-to-wall gold, with no empty space? If the cube is cut in half, the two pieces
Quark still retain their chemical identity as solid gold. But what if the pieces are cut again
composition and again, indefinitely? Will the smaller and smaller pieces always be gold? Ques-
of a proton
tions such as these can be traced back to early Greek philosophers. Two of them—
Proton Leucippus and his student Democritus—could not accept the idea that such cut-
tings could go on forever. They speculated that the process ultimately must end
Neutron
when it produces a particle that can no longer be cut. In Greek, atomosmeans “not
sliceable.” From this comes our English word atom.
Gold Let us review briefly what is known about the structure of matter. All ordinary
nucleus matter consists of atoms, and each atom is made up of electrons surrounding a
central nucleus. Following the discovery of the nucleus in 1911, the question
arose: Does it have structure? That is, is the nucleus a single particle or a collection
of particles? The exact composition of the nucleus is not known completely even
today, but by the early 1930s a model evolved that helped us understand how the
nucleus behaves. Specifically, scientists determined that occupying the nucleus are
two basic entities, protons and neutrons. The protoncarries a positive charge, and a
Nucleus
specific element is identified by the number of protons in its nucleus. This num-
ber is called the atomic numberof the element. For instance, the nucleus of a hy-
drogen atom contains one proton (and so the atomic number of hydrogen is 1),
the nucleus of a helium atom contains two protons (atomic number 2), and the
Gold
nucleus of a uranium atom contains 92 protons (atomic number 92). In addition
atoms
to atomic number, there is a second number characterizing atoms—mass num-
ber,defined as the number of protons plus neutrons in a nucleus. As we shall see,
the atomic number of an element never varies (i.e., the number of protons does
not vary) but the mass number can vary (i.e., the number of neutrons varies). Two
or more atoms of the same element having different mass numbers are isotopes
of one another.
The existence of neutrons was verified conclusively in 1932. A neutron has no
charge and a mass that is about equal to that of a proton. One of its primary pur-
poses is to act as a “glue” that holds the nucleus together. If neutrons were not
present in the nucleus, the repulsive force between the positively charged particles
Gold
cube would cause the nucleus to come apart.
But is this where the breaking down stops? Protons, neutrons, and a host of
other exotic particles are now known to be composed of six different varieties of
particles called quarks, which have been given the names of up, down, strange,
charm, bottom, and top. The up, charm, and top quarks have charges of (cid:4)2 that of
3
Figure 1.2 Levels of organization the proton, whereas the down, strange, and bottom quarks have charges of (cid:2)1
3
in matter. Ordinary matter consists that of the proton. The proton consists of two up quarks and one down quark
of atoms, and at the center of each
(Fig. 1.2), which you can easily show leads to the correct charge for the proton.
atom is a compact nucleus consist-
Likewise, the neutron consists of two down quarks and one up quark, giving a net
ing of protons and neutrons. Pro-
tons and neutrons are composed of charge of zero.
quarks. The quark composition of
a proton is shown.
1.3 Density 9
1.3 DENSITY
A property of any substance is its density (cid:5)(Greek letter rho), defined as the A table of the letters in the Greek
amount of mass contained in a unit volume, which we usually express as mass per alphabet is provided on the back
unit volume: endsheet of this textbook.
m
(cid:5)(cid:2) (1.1)
V
For example, aluminum has a density of 2.70 g/cm3, and lead has a density of
11.3 g/cm3. Therefore, a piece of aluminum of volume 10.0cm3 has a mass of
27.0g, whereas an equivalent volume of lead has a mass of 113g. A list of densities
for various substances is given Table 1.5.
The difference in density between aluminum and lead is due, in part, to their
different atomic masses.The atomic massof an element is the average mass of one
atom in a sample of the element that contains all the element’s isotopes, where the
relative amounts of isotopes are the same as the relative amounts found in nature.
The unit for atomic mass is the atomic mass unit (u), where 1 u(cid:6)1.660 540 2(cid:1)
10(cid:2)27kg. The atomic mass of lead is 207u, and that of aluminum is 27.0u. How-
ever, the ratio of atomic masses, 207u/27.0u(cid:6)7.67, does not correspond to the
ratio of densities, (11.3 g/cm3)/(2.70 g/cm3)(cid:6)4.19. The discrepancy is due to
the difference in atomic separations and atomic arrangements in the crystal struc-
ture of these two substances.
The mass of a nucleus is measured relative to the mass of the nucleus of the
carbon-12 isotope, often written as 12C. (This isotope of carbon has six protons
and six neutrons. Other carbon isotopes have six protons but different numbers of
neutrons.) Practically all of the mass of an atom is contained within the nucleus.
Because the atomic mass of 12C is defined to be exactly 12u, the proton and neu-
tron each have a mass of about 1u.
One mole (mol) of a substance is that amount of the substance that con-
tains as many particles (atoms, molecules, or other particles) as there are
atoms in 12g of the carbon-12 isotope. One mole of substance A contains the
same number of particles as there are in 1mol of any other substance B. For ex-
ample, 1mol of aluminum contains the same number of atoms as 1mol of lead.
TABLE 1.5 Densities of Various
Substances
Substance Density (cid:1)(103kg/m3)
Gold 19.3
Uranium 18.7
Lead 11.3
Copper 8.92
Iron 7.86
Aluminum 2.70
Magnesium 1.75
Water 1.00
Air 0.0012
10 CHAPTER 1 Physics and Measurements
Experiments have shown that this number, known as Avogadro’s number, N , is
A
N (cid:6)6.022 137(cid:1)1023 particles/mol
A
Avogadro’s number is defined so that 1mol of carbon-12 atoms has a mass of
exactly 12g. In general, the mass in 1mol of any element is the element’s atomic
mass expressed in grams. For example, 1mol of iron (atomic mass(cid:6)55.85u) has
a mass of 55.85g (we say its molar massis 55.85 g/mol), and 1mol of lead (atomic
mass(cid:6)207u) has a mass of 207g (its molar mass is 207 g/mol). Because there
are 6.02(cid:1)1023particles in 1mol of anyelement, the mass per atom for a given el-
ement is
molar mass
m (cid:6) (1.2)
atom N
A
For example, the mass of an iron atom is
55.85 g/mol
m (cid:6) (cid:6)9.28(cid:1)10(cid:2)23 g/atom
Fe 6.02(cid:1)1023 atoms/mol
EXAMPLE 1.1 How Many Atoms in the Cube?
A solid cube of aluminum (density 2.7 g/cm3) has a volume minum (27g) contains 6.02(cid:1)1023atoms:
of 0.20 cm3. How many aluminum atoms are contained in the
cube? NA (cid:6) N
27 g 0.54 g
Solution
Since density equals mass per unit volume, the 6.02(cid:1)1023 atoms N
mass mof the cube is (cid:6)
27 g 0.54 g
m(cid:6)(cid:5)V(cid:6)(2.7 g/cm3)(0.20 cm3)(cid:6)0.54 g (0.54 g)(6.02(cid:1)1023 atoms)
N(cid:6) (cid:6) 1.2(cid:1)1022 atoms
To find the number of atoms Nin this mass of aluminum, we 27 g
can set up a proportion using the fact that one mole of alu-
1.4 DIMENSIONAL ANALYSIS
The word dimension has a special meaning in physics. It usually denotes the physi-
cal nature of a quantity. Whether a distance is measured in the length unit feet or
the length unit meters, it is still a distance. We say the dimension—the physical
nature—of distance is length.
The symbols we use in this book to specify length, mass, and time are L, M,
and T, respectively. We shall often use brackets [ ] to denote the dimensions of a
physical quantity. For example, the symbol we use for speed in this book is v, and
in our notation the dimensions of speed are written [v](cid:6)L/T. As another exam-
ple, the dimensions of area, for which we use the symbol A, are [A](cid:6)L2. The di-
mensions of area, volume, speed, and acceleration are listed in Table 1.6.
In solving problems in physics, there is a useful and powerful procedure called
dimensional analysis. This procedure, which should always be used, will help mini-
mize the need for rote memorization of equations. Dimensional analysis makes
use of the fact that dimensions can be treated as algebraic quantities.That is,
quantities can be added or subtracted only if they have the same dimensions. Fur-
thermore, the terms on both sides of an equation must have the same dimensions.